Preuve de la conjecture de Poincaré en déformant la métrique par la courbure de Ricci

Besson, Gérard

Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 947, pp. 309-347.

Nous présentons la preuve de la conjecture de Poincaré, concernant les variétés compactes simplement connexes de dimension $3$, proposée par G. Perel’man. Elle repose sur l’étude de l’évolution de métriques riemanniennes sous le flot de la courbure de Ricci et sur les travaux antérieurs de R. Hamilton. Après une introduction aux techniques analytiques et géométriques développées par R. Hamilton, nous tentons de décrire la méthode de chirurgie métrique utilisée par G. Perel’man pour franchir les temps pour lesquels la courbure scalaire tend vers l’infini dans certaines parties de la variété. La preuve de la conjecture de Poincaré repose alors sur la preuve de l’extinction en temps fini du flot avec chirurgies, sous certaines hypothèses, que nous présentons dans la version élaborée par T. Colding et W. Minicozzi.

We present the proof of the Poincaré conjecture, on closed simply related three-manifolds, proposed by G. Perel'man. It relies on the study of riemannian metrics evoluting under the Ricci flow and on previous works by R. Hamilton. After and introduction to the analytical and geometrical techniques developped by R. Hamilton, we try to describe the technique of metric surgery used by G. Perel'man to go through the singular times for which the scalar curvature goes to infinity on certain parts of the manifold. the proof of the Poincaré conjecture then relies on the proof of the finite extinction time of the flow with surgeries, under certain assumptions, for which we present a version due to T. Colding and W. Minicozzi.

Classification : 57N10, 53C44, 58J35
Mot clés : variété de dimension $3$, conjecture de Poincaré, flot de Ricci
Keywords: three-manifolds, Poincaré conjecture, Ricci flow

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